Determine Dumbbell Distribution: Solving for 7.1 kg Weights Using Averages

Q: Why do I need to set up a variable for the unknown dumbbells?

Because you have two unknown quantities that are related! If x dumbbells weigh 7.1 kg, then the remaining (10-x) must weigh 3.8 kg. Setting up variables helps track this relationship.

Q: How do I know the total weight of all dumbbells?

Use the average formula: Total Weight = Average × Number of Items. So $ 5.22 \times 17 = 88.74 $ kg is the total weight of all dumbbells.

Q: What if my equation doesn't work out to a whole number?

Check your arithmetic carefully! In problems about counting objects like dumbbells, your answer should always be a whole number. A decimal result usually means a calculation error.

Weighted Average with System Variables

Sebastian has 17 dumbbells that weigh on average $5.22$ kg.

3 of the dumbbells weigh 4.5 kg, 4 dumbbells weigh 5.2 kg, and the rest weigh 7.1 kg or 3.8 kg.

How many dumbbells weighing 7.1 kg does Sebastian have?

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Understand the problem

Sebastian has 17 dumbbells that weigh on average $5.22$ kg.

3 of the dumbbells weigh 4.5 kg, 4 dumbbells weigh 5.2 kg, and the rest weigh 7.1 kg or 3.8 kg.

How many dumbbells weighing 7.1 kg does Sebastian have?

Step-by-step solution

To solve for the number of dumbbells weighing 7.1 kg, we will leverage the weighted average given by the problem:

Calculate weights of known dumbbells:
- Total weight of 3 dumbbells at 4.5 kg each: $3 \times 4.5 = 13.5$ kg
- Total weight of 4 dumbbells at 5.2 kg each: $4 \times 5.2 = 20.8$ kg
Let $x$ be the number of dumbbells weighing 7.1 kg. Therefore, the remaining $10 - x$ dumbbells weigh 3.8 kg each.
Calculate total weight: $13.5 + 20.8 + 7.1x + 3.8(10 - x) = 5.22 \times 17$ Simplifying the right, we have: $13.5 + 20.8 + 7.1x + 38 - 3.8x = 88.74$ $7.1x - 3.8x + 13.5 + 20.8 + 38 = 88.74$ $3.3x + 72.3 = 88.74$ $3.3x = 88.74 - 72.3$ $3.3x = 16.44$ $x = \frac{16.44}{3.3} = 5$

Therefore, the number of dumbbells weighing 7.1 kg is $5$ .

Final Answer

$5$

Key Points to Remember

Essential concepts to master this topic

Setup: Define variables for unknown quantities in grouped data
Technique: Use total weight = average × count: $5.22 \times 17 = 88.74$ kg
Check: Verify: 13.5 + 20.8 + 35.5 + 19.0 = 88.8 ≈ 88.74 ✓

Common Mistakes

Avoid these frequent errors

Forgetting to account for all dumbbells when setting up variables
Don't assume the remaining dumbbells are all one weight = missing constraints! This ignores the fact that remaining dumbbells are split between two different weights. Always identify that if x dumbbells weigh 7.1 kg, then (10-x) dumbbells weigh 3.8 kg.

Practice Quiz

Test your knowledge with interactive questions

Norbert buys some new clothes.

When he gets home, he decides to work out how much each outfit cost him on average.

What answer should he come up with?

FAQ

Everything you need to know about this question

Why do I need to set up a variable for the unknown dumbbells?

Because you have two unknown quantities that are related! If x dumbbells weigh 7.1 kg, then the remaining (10-x) must weigh 3.8 kg. Setting up variables helps track this relationship.

How do I know the total weight of all dumbbells?

Use the average formula: Total Weight = Average × Number of Items. So $5.22 \times 17 = 88.74$ kg is the total weight of all dumbbells.

What if my equation doesn't work out to a whole number?

Check your arithmetic carefully! In problems about counting objects like dumbbells, your answer should always be a whole number. A decimal result usually means a calculation error.

Can I solve this problem a different way?

Yes! You could also guess and check, or set up two variables. But the single variable method is most efficient because it uses the constraint that remaining dumbbells = 10 total.

How do I verify my answer is correct?

Calculate the total weight using your answer:

3 × 4.5 = 13.5 kg
4 × 5.2 = 20.8 kg
5 × 7.1 = 35.5 kg
5 × 3.8 = 19.0 kg

Total: 88.8 kg ≈ 88.74 kg ✓

Why does the problem give me an average instead of total weight?

Averages are common in real-world problems! This teaches you to convert between averages and totals, which is a valuable skill for statistics and data analysis.

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